Morphological descriptors — the mathematics: Gonzalez & Woods define shape descriptors as dimensionless ratios that characterize geometry independent of size. Compactness (also called circularity or form factor) is 4πA/P², which equals 1 for a perfect circle and decreases as shape deviates from circularity. This single number captures shape information that would require thousands of boundary coordinates to describe explicitly. The Feret ratio (minimum caliper diameter / maximum caliper diameter) captures elongation: 1 for a circle, approaching 0 for a line.
Why intensity ≠ concentration: Dobrucki and Pawley both emphasize this critical point: the fluorescence intensity measured at a pixel is not simply proportional to the local fluorophore concentration. Confounding factors include vignetting (non-uniform illumination across the field), focal plane position (above or below the plane of focus), spherical aberration (varies with depth), photobleaching (varies with exposure history), and quenching (varies with local chemical environment). Pawley notes that "voxel brightness is NOT simply proportional to fluorophore concentration." This means that comparing absolute intensities between cells is only valid when these confounding factors are controlled or corrected.
The principal axes: Eccentricity is derived from the eigenvalues of the pixel coordinate covariance matrix. The eigenvectors define the principal axes (longest and shortest dimensions) of the object, and the eigenvalue ratio quantifies elongation. When this ratio is near 1, the object is roughly circular; when much greater than 1, the object is elongated. This is the same mathematical framework as PCA (principal component analysis) applied to spatial coordinates rather than data features.
Population statistics vs. single-cell accuracy: Individual cell measurements are noisy — Poisson photon statistics, digitization effects, and boundary ambiguity all contribute uncertainty. But population statistics (mean intensity of 10,000 cells) are precise because errors average out across many cells. When interpreting measurements, distinguish between conclusions that rely on individual cell values (which need high SNR) and those that rely on population distributions (which are robust to per-cell noise).
Shape descriptors like compactness and eccentricity capture nuclear morphology in single numbers — a compactness of 0.9 means nearly circular, 0.4 means quite irregular. Intensity measurements seem straightforward but are affected by many factors beyond true biomarker concentration — illumination uniformity, focus position, and photobleaching all contribute. Population statistics (averaging over thousands of cells) are much more reliable than individual cell measurements because these errors cancel out.
